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I need help in this problem ,royden 4th edition . Let $g$ be integrable over $[a,b]$ .Define the antiderivative of $g$ to be the function $f$ defined on $[a,b]$ by $f(x)=\int_{a}^{x} g~~ for~all~ x\in [a,b]$. Show that $f$ is differentiable almost everywhere on $(a,b)$.

I know from Lebesgues Theorem if the function $f$ is monotone on the open intervals $(a,b)$ , then it is differentiable almost everywhere on $(a,b)$ But integrable not always monotone .

and I can not show that if $f$ is not differentiable at $x$ then $x \in E$ and $m(E)=0$ These are the only information I have about differentiable almost everywhere

I saw your comment but I can not add comment , how can I discusses your comment with you

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    $\begingroup$ For real-valued $g$, write $f$ as the difference of two monotonic functions. $g = \lvert g\rvert - (\lvert g\rvert - g)$ $\endgroup$ – Daniel Fischer Oct 28 '15 at 19:25
  • $\begingroup$ See the fundamental theorem of calculus, and also the well-known fact that an integrable function (in the Riemann sense) is almost everywhere continuous. $\endgroup$ – Arthur Oct 28 '15 at 19:29
  • $\begingroup$ @DanielFischer Why should $g$ equal such a difference? $\endgroup$ – zhw. Oct 28 '15 at 19:41
  • $\begingroup$ Does "integrable" mean Riemann integrable? $\endgroup$ – zhw. Oct 28 '15 at 19:41
  • $\begingroup$ Whether $g$ is Lebesgue integrable or Riemann integrable we can write $g=g_1-g_2$ where $g_1$ and $g_2$ are nonnegative and integrable in the same sense. If eva had known that functions of bounded variation are a.e. differentiable then no hint would be needed. $\endgroup$ – B. S. Thomson Oct 28 '15 at 19:55
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Based on the comments from @Daniel Fischer

Since $g = |{g}| - (|{g}|-g)$ and $g$ is integrable, the linearity of Lebesgue integral implies \begin{align*} f(x) = \int_a^x g = \int_a^x |{g}| - (|{g}|-g) = \int_a^x |{g}| - \int_a^x(|{g}|-g) = f_1(x)-f_2(x), \end{align*} where $f_1(x) = \int_a^x |{g}|$ and $f_2(x) = \int_a^x(|{g}|-g)$. Since $|{g}|\geq 0$ and $|{g}|-g\geq0$, the functions $f_1$ and $f_2$ are increasing on $(a,b)$. Thus, $f_1$ and $f_2$ are differentiable almost every on $(a,b)$. Since addition preserves differentiability, $f$ is differentiable a.e. on $(a,b)$.

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